If my daughter was born in a foreign country and doesn't yet have a SSN, can I still file my taxes? (It seems the SSN field is mandatory for her.)

You can either ignore the requirement to enter her ITIN, or you can enter a "dummy" SSN 305-44-0001 and then after you print your return you will need to mail it.  You cannot e-file your return.

In order to claim your daughter on your return, she would need to apply for an ITIN (Individual Taxpayer Identification Number) with  your return.

You will need to complete Form  W-7 and attach it to your return.  You cannot e-file.

Instructions for W-7

As the form asks for original documents for her, I would strongly suggest that you and your daughter go to an IRS Taxpayer Assistance Center once your return is finished and her W-7 form is completed.

In person. You can apply for your ITIN by visiting designated IRS Taxpayer Assistance Centers (TACs). They can verify original documentation and certified copies of the documentation from the issuing agency for primary and secondary applicants and their dependents. For dependents, TACs can verify passports, national identification cards, and birth certificates. These documents will be returned to you immediately. Service at TACs is by appointment only. Appointments can be scheduled by calling 844-545-5640.

See IRS.gov/W7DocumentVerification for a list of designated TACs that offer ITIN document authentication service. TACs that don’t offer ITIN document authentication service will mail the original documents, Form W-7, and the tax return to the IRS Austin Service Center for processing.